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The correct option is (b) \(\frac{\partial u}{\partial X}\)

Explanation: Here,

\(\frac{\partial}{\partial x}\VEC{i}+\frac{\partial}{\partial y}\vec{j}+\frac{\partial}{\partial z}\vec{K}\)

\(\vec{V}= u\vec{i} +v\vec{j}+W\vec{k}\)

∇.\(\vec{V}=\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}+\frac{\partial w}{\partial z}\)

For one-dimensional flow,

∇.\(\vec{V}=\frac{\partial u}{\partial x}\).

The CORRECT choice is (d) both the RATE of heat transferred to the system and the rate of heat generated by the system

To explain I WOULD SAY: Heat can be added to a system in two ways:

 Transfer of heat across the SURFACE of the element by surface forces.

 The heat generated by the system itself.

The CORRECT OPTION is (a) Velocity vector

Best explanation: The physical property in general transport equation should be REPLACED by velocity vector. This will result in momentum equation. For mass CONSERVATION, Φ = 1. For momentum (mass×velocity) conservation, Φ = 1×\(\vec{V}=\vec{V}\).

Correct OPTION is (a) Density and temperature

Easy explanation: There are FOUR thermodynamic VARIABLES – density, temperature, pressure and specific internal energy. Among these, pressure and specific internal energy can be REPRESENTED USING density and temperature.

The CORRECT OPTION is (c) ADIABATIC and inviscid flows

For explanation I would say: Euler equations constitute the governing equations of flow for adiabatic and inviscid flows. Here, the dissipative transport of flow PROPERTIES is neglected.

Correct option is (c) τxx, τyx, τzx

To explain I would SAY: τxx, τyx, τzx represent shear stresses DUE to VISCOUS effects; u is the x-velocity; fx is the body force and P is the PRESSURE. τxx,τyx,τzx should become zero for the flow to be in-viscid and the equations to be Eulerian.

The correct option is (b) The two terms are ALWAYS opposite in sign

The explanation: The “net mass FLOW ACROSS the fluid element” being positive means that it is outward flow. If flow is outward, mass inside the fluid element decreases leading to a NEGATIVE “rate of change of mass inside the element”. Thus, the two terms are always opposite in sign that they can be SUMMED up to get zero.

The CORRECT choice is (b) Temperature

The best I can explain: It is a MUST to solve the energy EQUATION when we WANT the temperature distribution of the system. The energy equation can give information about temperature.

Right option is (c) \(\frac{\partial\rho}{\partial t}(DX \,DY \,dz) \)

To explain: Mass=density × volume

mass of fluid element=ρ×dx dy dz

rate of CHANGE in mass of fluid element=\(\frac{\partial\rho}{\partial t}dx \,dy \,dz \)

Correct ANSWER is (B) \(\iiint_V\rho DV\)

To elaborate: Mass=density × volume

mass INSIDE dV=ρdV

mass inside \( V=\iiint_V\rho dV\).

Right option is (B) LOCAL DERIVATIVE, convective derivative

Explanation: SUBSTANTIAL derivative is the ADDITION of local derivative (based on fixed model) and convective derivative (based on motion of the model).

The CORRECT answer is (d) \(\iint_s(\VEC{V}\Delta t).\vec{ds}\)

To elaborate: The CHANGE in volume of the WHOLE control volume is the summation of \((\vec{V} \Delta t).\vec{ds}\) over the total control surface S. This summation becomes integral as \(\vec{ds}\) is elemental. Therefore, \(\iint_s(\vec{V}\Delta t).\vec{ds}\) is the total change.

The correct option is (d) Has the same particles ALWAYS inside it

To elaborate: As the finite control volume moves ALONG with the flow, the same particles are always CONCERNED. The mass inside this model also does not CHANGE.

The correct choice is (a) Partial differential equation

The best explanation: Infinitesimally small element MODELS directly LEADS to a partial differential equation. The element is infinitesimal in the sense of differential calculus.

Correct answer is (a) Volume integral term of control volume

To EXPLAIN: Using Leibniz rule, the DIFFERENTIATION of an integral term can be REDUCED. Here, differential of integral exists in the Volume integral term of Control Volume which is given by \(\frac{d}{dt}(\int_vb \rho dV).\)

The correct OPTION is (b) Both the LEFT-hand side and the right-hand side of the equation change

For explanation: Changes are applied to only left-hand side TERMS in the equations, but they affect both the left and right-hand sides of the equation.

Right CHOICE is (d) Thermal conductivity

Easy EXPLANATION: As the FLOW considered by Euler equations is ADIABATIC, heat cannot ENTER or exit the system. So, the thermal conduction is omitted.

Correct answer is (B) They increase entropy

To explain: Dissipation is the process where ENERGY is transformed from one form into ANOTHER. This TRANSFORMATION increases the entropy of the system.

Right answer is (d) GAUSS Divergence Theorem

The best I can EXPLAIN: The expansion of non-conservative integral EQUATION gives two volume integral terms. One of these terms representing the mass flow is converted into SURFACE integral using the Gauss Divergence theorem.

Right option is (c) \(\frac{D}{DT} \iiint_V\rho dV\)

EXPLANATION: Substantial DERIVATIVE is USED as the model is moving.

mass=density × volume

mass inside dV=ρdV

mass inside \( V=\iiint_V\rho dV\)

time RATE of change of mass inside \(\frac{D}{Dt} \iiint_V\rho dV\).

Correct option is (d) Heat conduction

For explanation I WOULD say: Diffusive heat TRANSFER is called heat conduction. This refers to the process of heat transfer WITHOUT any movement of the particles.

The correct answer is (c) Parcel’s centre of MASS at INITIAL time

To elaborate: Parcel’s centre of pressure is not suitable as it depends not only on position but on MANY variables too. INSTANTANEOUS time continuously varies along with the path. So, it cannot be a TAG.

Correct choice is (B) Thermodynamic EQUILIBRIUM

The explanation: Thermodynamic equilibrium is a state of a matter where there is no transfer of energy. This condition can be USED to relate the thermodynamic properties with one ANOTHER.

Correct choice is (c) u,v,w,T,p,ρ

For EXPLANATION: There are SIX DEPENDENT variables in the Navier-Stokes equations. They are PRESSURE (p), temperature (T), density (ρ) and three COMPONENTS of the velocity vector (u,v,w).

Right CHOICE is (c) Energy equation, mass and momentum EQUATIONS

For EXPLANATION I would say: Equations of the state provide the linkage of Energy equation with mass and momentum equations. They give the thermodynamic properties in terms of the state VARIABLES.

The correct choice is (a) DENSITY, PRESSURE, specific internal energy, temperature

Explanation: Velocity is a PROPERTY completely RELATED to FLUID flow. The other properties – density, pressure, specific internal energy and temperature are thermodynamic.

The CORRECT choice is (d) TIME rate of change at a fixed POINT

Best explanation: Local derivative is the term \(\frac{\PARTIAL}{\partial t}\) of a PROPERTY. This defines the time rate of change of a property at a particular point with the assumption that the point is fixed.

Correct option is (b) velocity field

To explain: Velocity field defines the MOTION of fluid. Parcels ALSO follow the same path of the fluid and THEREFORE, parcels can be SAID to follow the velocity field.

The correct answer is (c) equations of state

The BEST explanation: Equations of state give the RELATIONSHIP between temperature and internal ENERGY which is i=Cv T. Using this RELATION, one can obtain the energy equation in internal energy TERMS.

The CORRECT CHOICE is (c) VISCOUS FORCES

The best I can explain: Viscous forces are tensors. The other forces given here (Gravitational, viscous and ELECTROMAGNETIC forces) are vectors.

Right option is (c) \(a_x=\frac{DU}{DT}\)

Easy explanation: ACCELERATION is the rate of change of velocity. Acceleration in the x-direction is the rate of change of velocity in x-direction (u). As the MODEL is moving, rate of change is given by substantial derivative. Thus the acceleration in x-direction is \(a_x=\frac{Du}{Dt}\).

The correct OPTION is (b) \([(\vec{V}\Delta t).\vec{n}]dS\)

The explanation is: The CHANGE in CONTROL VOLUME can be calculated as the volume of a cylinder with altitude \((\vec{V}\Delta t).\vec{n}\) (product of velocity and time) and BASE area dS.

Correct option is (c) MOVING models

The best explanation: Divergence of velocity involves the CHANGE of VOLUME of the FLUID MODEL. For stationary models, volume does not change. So, this applies for only moving fluid elements.

The correct choice is (C) Fluids are not stationary and they have DIFFERENT VELOCITIES in different parts

Easiest explanation: A solid body can be easily divided into different parts. A fluid is squishy and it has different velocities in different parts. This creates the need for a MODEL to analyse it.

Correct choice is (a) momentum equation

To explain I would say: TOTAL energy is the sum of internal energy and kinetic energy. By some manipulation in the momentum equation, we can get the kinetic energy TERMS. If this is SUBTRACTED from the energy equation WRITTEN in total energy terms, we can get the same in terms of internal energy.